optimal nozzle shapes of co2-n2-h2o gasdynamic lasers

in an axisymmetric co2-n2-h2o gas dynamic laser, let ? denote the intersection of the vertical plane of symmetry with the upper part of the (supersonic) nozzle. to obtain a maximal small signal gain, some authors have tested several families of curves for ?. to find the most general solution for ?, an application of pontryagin’s principle led to the conjuncture that the optimal ? must consist of two straight lines of slopes m and 0 smoothly joined by a parabolic arc. (the parabolic section will vanish if nonsmooth ? is allowed.) the conjecture was settled in the affirmative for special cases. the present work extends these results in the following directions. (i) for the nonsmooth case, pontryagin’s principle produces no singularity and ? consists of k straight lines of certain slopes m and 0. (ii) a “semi-uncoupled” approximation is used to show, in (i), that k = 2. (“coupled” stands for the dynamic coupling between vibrational temperatures and translational temperature.) (iii) an uncoupled approximation is used in the smooth case to show that the general ? consists of a line segment of slope m, a parabolic arc and a horizontal line. (iv) the small signal gain increases whenever the slope m and/or the curvature of the parabolic section increase. however, the latter two quantities must be bounded to reduce gas detachment from the walls or oblique shock waves in the active media. (v) finally, the optimal shapes and gains are numerically calculated for several values of the stagnation pressure and molar fractions of the gas mixture.